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Planar quantum squeezing and atom interferometry

Qiang-You He, Shi-Guo Peng, P. D. Drummond, M. D. Reid

Published Jan 1, 2011• 3 citations
Publisher

Abstract

We obtain a novel planar squeezing uncertainty relation for spin variances in a plane, and show how to obtain such planar squeezed states using a BEC. These minimize interferometric phase-noise at all phase angles simultaneously.
DOI
10.1364/iqec.2011.i1110

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We obtain a lower bound on the sum of two orthogonal spin component variances in a plane. This gives a planar uncertainty relation which holds even when the Heisenberg relation is not useful. We investigate the asymptotic, large-JJJ limit and derive the properties of the planar quantum squeezed states that saturate this uncertainty relation. These states extend the concept of spin squeezing to any two conjugate spin directions. We show that planar quantum squeezing can be achieved experimentally as the ground state of a Bose-Einstein condensate in two coupled potential wells with a critical attractive interaction. These states reduce interferometric phase noise at all phase angles simultaneously. This is useful for one-shot interferometric phase measurements where the measured phase is completely unknown. Our results can also be used to derive entanglement criteria for multiple spins JJJ at separated sites, with applications in quantum information.
Reduction of quantum noise in one spin component is a significant tool for enhancing the sensitivities of interferometers and atomic clocks. It has been recently implemented for ultra-cold atomic Bose-Einstein condensate (BEC) interferometer. This type of quantum noise reduction reduces the measurement noise near some predetermined phase. However, if the phase is completely unknown prior to measurement, then it is not known which phase quadrature should be in a squeezed state. We introduce a novel planar squeezing uncertainty relation for spin variance in a plane, and analyze how to obtain such a planar quantum squeezed (PQS) state by using a double-well single component BEC, through the use of local nonlinear S-wave scattering interaction between trapped atoms. Here, we consider the PQS that is generated by using two hyperfine states in a two components BEC system, which is useful for quantum metrology. By comparison with the case of two spatial wells, the Hamiltonian parameters can be controlled in a more efficient way. The spin component can be measured by detecting the occupation number difference between the two internal modes, while one needs to observe a spatial interference pattern in the double well BEC case. This is the major difference between the internal and external cases. Another difference is that one can use the Rabi frequency Ω instead of the Josephson parameters to switch the Hamiltonian parameters through using a diabatic technique. Therefore the coupling could be switched off or on to study the different evolutions. PQS simultaneously reduces the quantum noises of two orthogonal spin projections below the standard quantum limit, while increases the noise in the third dimension. This allows the improvement in phase measurement at any phase-angle. PQS states that reductions of fluctuations everywhere in a plane have potential utility in "one-shot" phase measurement, where iterative or repeated measurement strategies cannot be utilized. The improved interferometric phase measurements and planar uncertainty relations are useful for detecting the entanglement in mesoscopic system between two distinguished modes regardless of the third component.
We derive a phase-entanglement criterion for two bosonic modes which is immune to number fluc- tuations, using the generalized Moore-Penrose inverse to normalize the phase-quadrature operator. We also obtain a phase-squeezing criterion that is immune to number fluctuations using similar techniques. These are utilized to obtain an operational definition of relative phase-measurement sensitivity, via analysis of phase measurement in interferometry. We show that these measures are proportional to enhanced phase-measurement sensitivity. The phase-entanglement criterion is a hallmark for a new type of quantum squeezing, namely planar quantum squeezing. This has the property that it squeezes two orthogonal spin directions simultaneously, which is possible owing to the fact that the SU(2) group that describes spin symmetry has a three-dimensional parameter space, of higher dimension than the group for photonic quadratures. The practical advantage of planar quantum squeezing is that, unlike conventional spin-squeezing, it allows noise reduction over all phase-angles simultaneously. The application of this type of squeezing is to quantum measure- ment of an unknown phase. We show that a completely unknown phase requires two orthogonal measurements, and that with planar quantum squeezing it is possible to reduce the measurement uncertainty independently of the unknown phase value. This is a different type of squeezing to the usual spin-squeezing interferometric criterion, which is only applicable when the measured phase is already known to a good approximation, or can be measured iteratively. As an example, we calcu- late the phase-entanglement of the ground state of a two-well, coupled Bose-Einstein condensate, similar to recent experiments. This system demonstrates planar squeezing in both the attractive and repulsive interaction regimes.
We derive a phase-entanglement criterion for two bosonic modes which is immune to number fluc- tuations, using the generalized Moore-Penrose inverse to normalize the phase-quadrature operator. We also obtain a phase-squeezing criterion that is immune to number fluctuations using similar techniques. These are utilized to obtain an operational definition of relative phase-measurement sensitivity, via analysis of phase measurement in interferometry. We show that these measures are proportional to enhanced phase-measurement sensitivity. The phase-entanglement criterion is a hallmark for a new type of quantum squeezing, namely planar quantum squeezing. This has the property that it squeezes two orthogonal spin directions simultaneously, which is possible owing to the fact that the SU(2) group that describes spin symmetry has a three-dimensional parameter space, of higher dimension than the group for photonic quadratures. The practical advantage of planar quantum squeezing is that, unlike conventional spin-squeezing, it allows noise reduction over all phase-angles simultaneously. The application of this type of squeezing is to quantum measure- ment of an unknown phase. We show that a completely unknown phase requires two orthogonal measurements, and that with planar quantum squeezing it is possible to reduce the measurement uncertainty independently of the unknown phase value. This is a different type of squeezing to the usual spin-squeezing interferometric criterion, which is only applicable when the measured phase is already known to a good approximation, or can be measured iteratively. As an example, we calcu- late the phase-entanglement of the ground state of a two-well, coupled Bose-Einstein condensate, similar to recent experiments. This system demonstrates planar squeezing in both the attractive and repulsive interaction regimes.